Browder Spectral Systems

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 103, Number 2, June 1988 BROWDER SPECTRAL SYSTEMS RAÚL E. CURTO AND A. T. DASH (Communicated by Paul S. Muhly) ABSTRACT. For two spectral systems cri and 02 on a Banach space 3?, the associated Browder spectral system is r&;i,2 : o i U r2. We prove that cT(,;12 possesses the projection and spectral mapping properties whenever o\ and 02 do (and satisfy a few additional mild assumptions). We also calculate cT( ;i,2 for tensor products. The results extend several previous works on Browder spectra. 1. Introduction. The Browder spectrum of an operator T acting on a Banach space 3f is usually defined as cr6(T): (7e(T)UcT(T)', the union of the essential spectrum and the limit points of the spectrum. B. Gramsch and D. Lay showed in [9] that b(f(T)) f(crb(T)) for every function / analytic on a neighborhood of a(T). Browder spectra have been considered by many authors; we only mention here the works [1. 6. 13, and 14], which deal with various notions of joint Browder spectra. We can encompass these notions in a very general one involving arbitrary spectral systems. Roughly speaking, a spectral system on 3f assigns a compact nonempty subset of Cn to every commuting n-tuple of operators on 3f (the sizes of the tuples are allowed to vary). Given two spectral systems tri and a2, the Browder spectral system associated with er, and a2 is &b;l,2 ' Oí U02, where ' stands for the set of limits points. We first obtain conditions for tTö;i,2 to possess the projection property and we subsequently proceed to consider the spectral mapping property via the functional representation obtained by W. Zelazko in [18] (see also [4, §§2 and 3]). Finally, we calculate 7ö;i,2in the case of tensor products for suitable choices of 7i and cr2, particularly o i o-pe and a2 op, the Taylor essential and Taylor spectra, respectively. Our main result, Theorem Received by the editors February 17, 1987 and, in revised form, May 4, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 47A10, 47A53, 47A50; Secondary 47A62. Key words and phrases. Spectral system, Browder joint spectra, projection products. Research of the first author partially supported by a grant from NSF. Research of the second author partially supported by a grant from NSERC. 1988 American 0002-9939/88 407 property, Mathematical tensor Society 1.00 .25 per page

R. E. CURTO AND A. T. DASH 408 2.8, establishes that whenever ai and a2 possess the projection property and every isolated point of o2(S) is isolated in o p(S), we have Ps(?b;l,2(S,T) T6;li2(S) for all T such that (S,T) is commuting. (Here S and T are tuples and Ps projects onto the S-coordinates; to be precise, a few extra assumptions are needed, but they are quite mild and satisfied by most known spectral systems.) Once we know that o-0-ii2 possesses the projection property, we can establish a spectral mapping theorem for (vector-valued) functions analytic in neighborhoods of the Taylor spectrum, extending to n-variables some of Gramsch and Lay's results. Theorem 2.8 also provides a good source of spectral systems possessing the projection property; as is known, such systems generally possess the spectral mapping property for polynomial mappings (see §3 below). 2. The projection property for Browder spectral systems. For 3? a Banach space, we let 5C(3f) denote the algebra of all (bounded) operators on 3f : f(3f) coL will denote the collection of all commuting n-tuples of elements in 5?(3f). FollowingZelazko[18], we write S c 2'(3f) if S G 5?(3f)c2L for some n 1. A spectral system â on 3f is an assignment S — à(S) defined on Un 1 ?(3?)coL and such that d(S) is a compact nonempty subset of Cn whenever S G ¿2?(3f )cc)m. A spectral system possesses the projection property if Psä(S, T) ¿ (5) and PTà(S, T) â(T) for all (S, T) C 2C(3f). (Here Ps and Pp are the canonical projections onto the coordinates of S and T, respectively.) Spectral systems possessing the projection property have a privileged position among general spectral systems: Under the assumption à Ç ¿ (the polynomially convex spectrum), such a à will also possess the spectral mapping theorem for functions analytic in a neighborhood of ¿ . (A more general result is true, but we will not try to describe it here; see [4, 11].) Our aim is to obtain the projection property for Browder spectral systems and then to apply the above result to give very general spectral mapping theorems for Browder spectra, extending greatly on previous work of B. Gramsch and D. Lay [9]. DEFINITION 2.1. Let 3f be a Banach space and let cri and a2 be two spectral systems on 3f. The Browder spectral system associated with oi and a2 is b;l,2 o i Ua2, where ' stands for the set of limit points. It goes without saying that 76;i,2is indeed a spectral system. Also, if ai ope and a2 aT (the Taylor essential and Taylor spectra, respectively, [4, 15]), and S is an operator on 3f, then o-b-i,2(S) ab(S), the usual Browder spectrum. PROPOSITION 2.2. Letoi anda2 be spectral systems for Jz?'(3f), and let o'b;i,t be the Browder spectral system associated possess the projection property. For (S,T) to 7i and o 2. Assume that 7i and a2 C 3\3f) we then have o-b-i,2(S) C Pscrb;h2(S,T). PROOF. Let A G erb;1?2(S). If A e cri(S), then there exists p G tri(T) such that (X,p) G (7i(S,T) Ç (rfc;i,2(S,T), and therefore A G Psob.i,2(S,T). a2(S)' instead, then A lim„ A„, where {A„} is a sequence of distinct If A points of

BROWDER SPECTRAL SYSTEMS 409 cr2(S). By the projection property for a2, we can then find pn G cr2(T) such that (Xn,pn) G a2(S,T). By the compactness of a2(T), there exists a subsequence pnic — p G ct2(T). Thus, (X,p) is the limit of a sequence of distinct points of a2(S,T), so \ G Pso-b.h2(S,T). D REMARK 2.3. The containment in Proposition 2.2 is usually the harder part in a proof of the projection property, since most spectral systems r satisfy â(S,T) Ç â(S) x ¿r(T); this is not the case here, as the following example shows. As usual, er/ and 7r denote the left and right spectra, respectively. EXAMPLE2.4. Let %? be a Hubert space and let S C 5f(3f) be such that 0 o-r(S) and cri(S) {0}UA", where K is compact and nonempty (one such S was constructed in [8]; see also [3]), and let T be an operator on %f such that 0Gct,(T)'. Consider (S I,I oi(S I,I T) T). By [10], at(S) x ai(T) [{0} x a¡(T)] Ú [K x tr,(T)]. It follows that 0 G Ti(S 1,1 T)', while 0 cri(S)'. If we let ax : ar and o 2: a¡, we see that o-b-i,2(S I,I T) lt6;1i2(5 I) x ct6;i,2(/ T). The above example indicates that additional assumptions on r2 must be made to obtain the reverse inclusion for the projection property. For a Banach space 3f we shall let 9pC\oo( ') denote the collection of infinite dimensional subspaces of 3? that admit a Banach space complement, i.e., * G 5*c\oo(3?) if and only if dim # oo and there exists a subspace yV such that 3f * JV'. Of course, if 3? is infinite dimensional, 3? G c;oo( )DEFINITION 2.5. Let à {¿ j e . r ) be a family of spectral systems (¿reacting on -#). We shall say that à is monotone if for all * G 9í'c.t00(3f) and all T G f(3f) such that TJT Ç Jt and TJf Ç jf (where Jf JT 3f), one has jf(T\.-it) Ç 7t(T). Most known spectral systems give rise to montone families. For instance, spatial systems (o i,o r,op, Slodkowski's joint spectra [13]), their essential counterparts (a¡e,o-e, etc.), and the product spectrum 7n : r x a x x cr) are monotone. Moreover, a calculation shows that the commutant and bicommutant spectra are monotone. Finally, we can see that à is monotone as follows: If T C 5C(3f), JÍ G S%:0O(äf), and TJt Ç Jf and 7VT C jV (where Jt JT thenaT(TU) Ç crT(T),so that &(TU) MTUOP M?X 3f), ¿CH,where denotes polynomially convex hull (recall that (op) è by [15, Theorem 5.2]). A similar argument works for the rationally convex spectrum, using [4, Application 3.9] instead. PROPOSITION 2.6. Let cri and a2 be spectral systems on 3f, (i) o i gives rise to a monotone family {(o i). } c.i¡c); (ii) \p\\jt Q {onU for all* G c;00(3f); (iii) ai(S,T) C o-i(S) x cr,(T) (all (S,T) C &(3f)); (iv) cT2(S,T)çz T2(S)xa2(T) (all (S,T) c S?(3?)). Then, if (v) isol.(T2(S) Ç isol. 7r(S), we have (*) PS ;1,2(5,T)Ç(70;1,2(S) and assume that

R. E. CURTO AND A. T. DASH 410 for allT such that (S,T) cS?(3f) and a¡(S,T) nar(S,T) Çai(S,T). (Here isol. denotes the set of isolated points.) REMARK 2.7. Assumptions (i)-(iv) are very mild and satisfied by most spectral systems. The important condition is (v); as we saw before (Example 2.4), not every a2 can produce a Browder spectral system satisfying (*). We shall see later that (v) holds in many instances. PROOF OF PROPOSITION 2.6. Let (\,p) G ab;i¿(S,T) and assume that A o b;i,2(S), i.e., A o-i(S) and A is not a limit point of a2(S). From (iii), (X,p) ai(S,T); thus, (A,//) G ct2(S,T)' and by (iv) we get at once that A G a2(S). Therefore, A G o2(S)\a2(S)', so that A is in isol.(72(S) Ç isol. rT(S). Let K : ({A} x cTT(T))r\aT(S,T) and L : aT(S,T)\K. [15, Theorem 4.9] there exists a decomposition Clearly aT(S,T) KÙL and by 3f * JV, where * and Jf are invariant for (S,T) and o-T((S,T)[ ) K, aT((S,T)\jr) L. It follows that t(S\#) — {A}, so that Tn(S\jr) {A}. \f* is infinite dimensional, we have * G fc.,oo{&) ander, (SU) {A} (sinceby (ii), cn(SU) C {A} and also (S ) ¿ 0). Moreover, o i(S\#) Ç tr,(S) by monotonicity and thus A G oi(S), a contradiction. Therefore * must be finite dimensional. Then {v: (\,v) G ap(S,T)} is finite, so that (X,p) G isol.aT(S,T). By another application of [15, Theorem 4.9], (X,p) G T¡(S,T) n o-r(S,T) (see the end of the proof of Theorem 2.11 below) and thus (X,p) G ai(S,T). Therefore, A G o i(S) (by (iii)), a contradiction. The proof of the proposition is now complete. THEOREM 2.8. Letoi anda2 be two spectral systems possessing the projection property. Assume that o i satisfies (i) and (ii) of Proposition 2.6. Then, if S C 2f(3?) and isol. o2(S) Ç isol. op(S), we have Pso-b;i,2(S,T) ab,i 2(S) for all T such that (S,T) C 2'(3f). PROOF. Combine Proposition 2.2 and Proposition 2.6 (with its proof). D COROLLARY 2.9. Let ci be a spectral system possessing the projection property and satisfying (i) and (ii) of Proposition 2.6. Then crbti,p : oi Ua'T possesses the projection property. COROLLARY 2.10. ab : ape U a'T possesses the projection property. We shall see now that 72can be quite general and still satisfy (iv) and (v) of Proposition 2.6. THEOREM 2.11. Let S C ¿2?(3f) and let ⧠be a commutative unital subalgebra of¿f(3f) containing S. Let A be an isolated point ofa (S). Then A is an isolated point of o t(S) and, a fortiori, an isolated point of cri(S) C\ar(S). PROOF. We know that cr (S) {A} ÙK, where K is compact. Assume that A g trT(5); since aT(S) Ç a. (S) we must have aT(S) Ç K. Let P f(S) G .93 be the idempotent associated with A constructed via the Shilov-Arens-Calderón- Waelbroeck functional calculus. Then 7 (P) f(o (S)) {0,1}, while ap(P) — f(crT(S)) {0}. Since r. (P) C â(P) and â(P) [ap(P)] we get a contradiction. Thus, A G iTp(S). The statement about a¡ n or follows from the fact that if tTT(S) {A}Ù L (L Ç K), then 3? * yV with aT{S\jr) {A} [15, Theorem